rated stability - lee.uerj.brvss2020/homepage-summer-school/lectures/vss-2019... · (erugin 1951,...

Rated Stability

L. Fridman

UNAM

L. Fridman Stability 1 / 34

Outline

1 IntroductionHistorical RemarksModel of Dynamical System

2 Stability NotionsUnrated StabilityRated StabilityNon-Asymptotic Stability


A.M. Lyapunov (1857-1918) and the first page of his thesis


Dynamic Systems and Stability

Pendulum Equation

Consider the pendulum equation

θ(t) + k θ(t) +g

rsin (θ(t)) = 0

where

θ – an inclination angle,

k – a friction coefficient

r – a length of pendulum,

g – the gravitationconstant.


Outline




Historical Remarks

Unrated Stability

Lyapunov Stability, Asymptotic Stability(Lyapunov 1892, Zubov 1957, Krasovskii 1959, LaSalle & Lefschetz 1960,Hahn 1961, Roxin 1965 etc)

Rated Stability

Exponential, Finite-time and Fixed-time Stability(Erugin 1951, Zubov 1957, Hahn 1961, Roxin 1966, Demidovich 1974,Bhat & Bernstein 2000, Orlov 2005, Levant 2005, Moulay & Perruquetti2005, Andrieu et al 2008,Cuz, Moreno, Fridman 2010, Polyakov 2012,...)


Outline




System Description

Model of the System

Consider the differential inclusion

x(t) ∈ F (t, x(t)), t ∈ R; (Sys)

x(t0) = x0, x0 ∈ R (IC)

Assumption

0 ∈ F (t, 0) for t ∈ R

Notation

Φ(t, t0, x0)– Set of all solutions of the Cauchy problem (Sys);

x(t, t0, x0) ∈ Φ(t, t0, x0)– a solution of (Sys)-(IC).


Weak Stability

Example

Weakly stable system{x1 = x2

x2 ∈ −(k1sign[x1] u k2sign[x2]

),, xi ∈ R,

2 cases

If k1 > k2 > 0 → x1 = 0, x2 = 0 is finite stable equilibrium point

If k2 > |k1| → x1(t) = constant, x2 = 0 is a solution.


Outline




Lyapunov Stability

Definition (Stability, Lyapunov 1892)

The origin of the system (Sys) is said to be Lyapunov stable if ∀ε ∈ R+

and ∀t0 ∈ R : ∃δ = δ(ε, t0) ∈ R+ such that ∀x0 ∈ B(δ)

1 any solution x(t, t0, x0) of Cauchy problem (Sys)-(IC) exists fort > t0;

2 x(t, t0, x0) ∈ B(ε) for t > t0.


Uniform Stability and Instability

Definition (Uniform Lyapunov Stability)

If the function δ in Definition of Lyapunov Stability does not depend on t0

then the origin is called uniformly Lyapunov stable.

If x ∈ F (x) is Lyapunov stable, then it is uniform Lyapunov stable

Proposition

If the origin of the system (Sys) is Lyapunov stable then x(t) = 0 is theunique solution of Cauchy problem (Sys)-(IC) with x0 = 0 and t0 ∈ R.

Definition (Instability)

The origin, which does not satisfy any condition from Lyapunov Stabilitydefinition, is called unstable.


Example (Uniformly Lyapunov Stable System)

{x1 ∈ sign[−x2],x2 ∈ sign[x1]

, x1, x2 ∈ R



No sliding motion{x1= sign[−x2],x2= sign[x1]

, x1, x2 ∈ R




, x1, x2 ∈ R

V = |x1|+ |x2|




, x1, x2 ∈ R

V = |x1|+ |x2|

V = sign(x1)x1 + sign(x2)x2 = 0




, x1, x2 ∈ R

V = |x1|+ |x2|

V = sign(x1)x1 + sign(x2)x2 = 0

Lyapunov Stability


Definition (Asymptotic attrativity)

The origin of the system (Sys) is said to be asymptotically attractive if∀t0 ∈ R exists a set U(t0) ⊆ Rn : U(t0) \ 0 is non-empty and ∀x0 ∈ U(t0)

any solution x(t, t0, x0) of Cauchy problem (Sys)-(IC) exists fort > t0;

limt→+∞

‖x(t, t0, x0)‖ = 0.

The set U(t0) is called attraction domain.


Definition (Asymptotic stability)

The origin of the system (Sys) is said to be asymptotically stable if it is

Lyapunov stable;

asymptotically attractive with an attraction domain U(t0) ⊆ Rn suchthat 0 ∈ int(U(t0)) for all t0 ∈ R.


Asymptotic attractivity may not imply Asymptotic Stability

Example (R.E. Vinograd 1957)

x1 =x2

1 (x2 − x1) + x52(

x21 + x2

2

) (1 +

(x2

1 + x22

)2)

and

x2 =x2

2 (x2 − 2x1)(x2

1 + x22

) (1 +

(x2

1 + x22

)2)


Definition (Uniform Asymptotic Attractivity)

The origin of the system (Sys) is said to be uniformly asymptoticallyattractive

if it is asymptotically attractive with a time-invariant attractiondomain U ⊆ Rn;

∀R ∈ R+, ∀ε ∈ R+ there exists T = T (R, ε) ∈ R+ such that theinclusions x0 ∈ B(R) ∩ U and t0 ∈ R imply x(t, t0, x0) ∈ B(ε) fort > t0 + T .


Uniform Asymptotic Stability

Definition (Uniform asymptotic stability)

The origin of the system (Sys) is said to be uniformly asymptoticallystable if it is uniformly Lyapunov stable and uniformly asymptoticallyattractive with an attraction domain U ⊆ Rn : 0 ∈ int(U).

Proposition (Clarke, Ledyaev, Stern 1998)

Let a set-valued function F : Rn → Rn be defined andupper-semicontinuous in Rn. Let F (x) be nonempty, compact and convexfor any x ∈ Rn.If the origin of the system

x ∈ F (x)

is asymptotically stable then it is uniformly asymptotically stable.


Outline




Definition (Exponential stability)

The origin of the system (Sys) is said to be exponentially stable if thereexist an attraction domain U ⊆ Rn : 0 ∈ int(U) and numbers C , r ∈ R+

such that‖x(t, t0, x0)‖ ≤ C‖x0‖e−r(t−t0), t > t0.

for t0 ∈ R and x0 ∈ U.

The theory of Linear Control Systems deals with exponential stability


Example (Linear Variable Structure System)

x = −(2− sign[sin(x)])x , x ∈ R, x(t0) = x0

‖x(t, t0, x0‖ ≤ |x0|e−(t−t0), t > t0

4 6 8 10

0

0.5

1

t

‖x(t,t 0,x0‖

‖x(t, 4, 1)‖e−(t−4)


Example (Homogeneous system)

x = −bxeα; x(0) = x0 0 < α < 1 1

1b·eα = | · |α sign[·]L. Fridman Stability 22 / 34


x = −bxeα; x(0) = x0 0 < α < 1 1

x0 >= 0

⇒ x(t) >= 0 ∀t ∈ [0, t1]

x = −xα

⇒ x(t) =(x1−α

0 − (1− α)t) 1

1−α

x(t) = 0 at t =x1−α

0

1− α



x = −bxeα; x(0) = x0 0 < α < 1 1

x0 >= 0

⇒ x(t) >= 0 ∀t ∈ [0, t1]

x = −xα

⇒ x(t) =(x1−α

0 − (1− α)t) 1

1−α

x(t) = 0 at t =x1−α

0

1− α

x0 <= 0

⇒ x(t) <= 0 ∀t ∈ [0, t1]

x = (−x)α

⇒ x(t) =((−x0)1−α − (1− α)t

) 11−α

x(t) = 0 at t =(−x0)1−α

1− α



x = −bxeα; x(0) = x0 0 < α < 1 1

x0 >= 0

⇒ x(t) >= 0 ∀t ∈ [0, t1]

x = −xα

⇒ x(t) =(x1−α

0 − (1− α)t) 1

1−α

x(t) = 0 at t =x1−α

0

1− α

x0 <= 0

⇒ x(t) <= 0 ∀t ∈ [0, t1]

x = (−x)α

⇒ x(t) =((−x0)1−α − (1− α)t

) 11−α

x(t) = 0 at t =(−x0)1−α

1− α

∴ x(t) = 0 ∀t ≥ |x0|1−α

1− α


Mechanical Example

Example (Deceleration of a Cart)


Mechanical Example


mx = −kd sign[x ]

m – mass

x – position

kd – coefficients of dryfriction


Mechanical Example


mx = −kd sign[x ]

m – mass

x – position


x(t) = 0, ∀t ≥ m

kd|x(0)|


Mechanical Example


0 1 2 3 4 5

0

2

4

t

x

mx = −kd sign[x ]

m – mass

x – position


x(t) = 0, ∀t ≥ m

kd|x(0)|


Outline




Finite-time attractivity

Introduce the functional T0 : Wn[t0,+∞) → R+ ∪ {0} by

T0(y(·)) = infτ≥t0:y(τ)=0

τ.

If y(τ) 6= 0 for all t ∈ [t0,+∞) then T0(y(·)) = +∞.Define the settling-time function of the system (Sys) as

T (t0, x0) = supx(t,t0,x0)∈Φ(t0,x0)

T0(x(t, t0, x0))− t0.

Definition (Finite-time attractivity)

The origin of the system (Sys) is said to be finite-time attractive if ∀t0 ∈ R existsa set V(t0) ⊆ Rn : V (t0) \ {0} is non-empty and ∀x0 ∈ V(t0)

any solution x(t, t0, x0) of Cauchy problem (Sys)-(IC) exists for t > t0;

T (t0, x0) < +∞ for x0 ∈ V(t0) and for t0 ∈ R.

The set V(t0) is called finite-time attraction domain.


Finite-time Stability (Erugin 1991, Zubov 1957, etc)

Definition (Finite-time stability, Roxin 1966)

The origin of the system (Sys) is said to be finite-time stable if it isLyapunov stable and finite-time attractive with an attraction domainV(t0) ⊆ Rn such that 0 ∈ int(V(t0)) for any t0 ∈ R.

Proposition (Bhat & Bernstein 2000)

If the origin of the system (Sys) is finite-time stable then it isasymptotically stable and x(t, t0, x0) = 0 for t > t0 + T0(t0, x0).


Uniform Finite-time Stability (Roxin 1966, Praly 1997, etc)

Definition (Uniform finite-time attractivity)

The origin of the system (Sys) is said to be uniformly finite-time attractiveif

it is finite-time attractive with a time-invariant attraction domainV ⊆ Rn;

T (t0, x0) is locally bounded on R× V uniformly on t0 ∈ R, i.e.

∀y ∈ V : ∃ε ∈ R+ ⇒ supt0∈R,

x0∈{y}uB(ε)⊂V

T (t0, x0) < +∞.


Uniform Finite-time Stability (Roxin 1966, Praly 1997, etc)

Definition (Uniform finite-time stability (Orlov 2005)))

The origin of the system (Sys) is said to be uniformly finite-time stable ifit is uniformly Lyapunov stable and uniformly finite-time attractive with anattraction domain V ⊆ Rn : 0 ∈ int(V).

Example

x ∈ −sign[x ], x ∈ R, T (t0, x0) = |x0|


Time invariance does not imply uniformity of FTS

Example (S.P. Bhat & D. Bernstein 2000)

Denote x i0 =(0, −1

i

)T, i = 1, 2, 3, . . .

x i0 → 0 and T (x i0)→ +∞


Example

Two uniformly finite-time stable systems Consider two systems2

(I ) x = −bxe12 (1− |x |) ,

(II ) x =

{−bxe

12 for x < 1,

0 for x ≥ 1,

which are uniformly finite-time stable with V = B(1).

T(I )(x0) = ln

(1 + |x0|

12

1− |x0|12

)T(I )(x0)→ +∞ if x0 → ±1

T(II )(x0) = 2|x0|12 .

T(II )(x0)→ 2 if x0 → ±1

2bxe[ρ] = |x |ρ sign[x ]L. Fridman Stability 29 / 34

Mechanical Example



Mechanical Example


mx = −(kd + kv x

2)

sign[x ], t > 0

m – mass

x – position

kd , kv – coefficients of dry andviscous friction


Mechanical Example


mx = −(kd + kv x

2)

sign[x ], t > 0

m – mass

x – position


x(t) = 0, ∀t ≥ Tmax =mπ

2√kdkv

and for any (x0, x0) ∈ R2


Mechanical Example


0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

t

x

x0 = 1 x0 = 10 x0 = 100 x0 = 1000 Tmax mx = −(kd + kv x

2)

sign[x ], t > 0

m – mass

x – position


x(t) = 0, ∀t ≥ Tmax =mπ

2√kdkv

and for any (x0, x0) ∈ R2


Example

x = −bxe12 − bxe

32 , x(0) = x0 > 0

dx√x (1 + x)

= −dt

z =√x ⇒ x = z2 dx = 2zdz

2

∫dz

1 + z2= −

∫dt

2 arctan√x = C − t, t = 0 ⇒ C = 2 arctan

√x0

2 arctan√x = 2 arctan

√x0 − t, ∀x0, t < π


Example

x = −√x −√x3

dx√x (1 + x)

= −dt

z =√x ⇒ x = z2 dx = 2zdz

2

∫dz

1 + z2= −

∫dt

2 arctan√x = C − t, t = 0 ⇒ C = 2 arctan

√x0

2 arctan√x = 2 arctan

√x0 − t, ∀x0, t < π


Fixed-time Stability(Balakrishan 1996, Andrieu et al2008,Cruz, Moreno, Fridman 2010,...)

Definition (Fixed-time attractivity)

The origin of the system (Sys) is said to be fixed-time attractive if

it is uniformly finite-time attractive with an attraction domain V;

T (t0, x0) is bounded on R× V, i.e.

∃Tmax ∈ R+ such that T (t0, x0) ≤ Tmax if t0 ∈ R, x0 ∈ V

Definition (Fixed-time stability, Polyakov 2012)

The origin of the system (Sys) is said to be fixed-time stable if it isLyapunov stable and fixed-time attractive with an attraction domainV ⊆ Rn : 0 ∈ int(V).


Fixed-time stability ⇒ NON-Asymptotic estimation &control

{x(t) = u(t)x(t) = x0,

x , u ∈ R



{x(t) = u(t)x(t) = x0,

x , u ∈ R

Asymptotic stabilisation:

u(t) = −x(t)

x(t) = e−tx0 → 0 if t → +∞



{x(t) = u(t)x(t) = x0,

x , u ∈ R

Finite-Time stabilisation:

u(t) = −bx(t)e0

x(t) = 0 for t ≥ ‖x0‖



{x(t) = u(t)x(t) = x0,

x , u ∈ R

Fixed-Time stabilisation:

u(t) = −bx(t)e12 − bx(t)e

32

x(t) = 0 for t ≥ π

independently of x0


Summary

In general case, attractivity does not imply stability.

Strong stability is more preferable for control applications.

Control theory is mainly interested in rated stability.


rated stability - lee.uerj.brvss2020/homepage-summer-school/lectures/vss-2019... · (erugin 1951,...

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